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We address the problem of conformally mapping the unit disk to polygons with elongations. The elongations cause the derivative of the conformal map to be exponentially large in some regions. This crowding phenomenon creates difficulties in standard numerical methods for the computation of the conformal map. We make use of the Schwarz-Christoffel representation of the mapping and show that a simple change to the existing algorithms introduced by Trefethen [SIAM J. Sci. Statist. Comput., 1 (1980), pp. 82-102] makes it feasible to accurately compute conformal maps to polygons even in the presence of extreme crowding. For an efficient algorithm it is essential that a good initial guess for the parameters of the Schwarz-Christoffel map be available. A uniformly close initial guess can be obtained from the cross-ratios of certain quadrilaterals, as introduced in the CRDT algorithm of Driscoll and Vavasis [SIAM J. Sci. Comput., 19 (1998), pp. 1783-1803]. We present numerical experiments and compare our algorithms with the CRDT which has been particularly designed to combat crowding.

Revisiting the Crowding Phenomenon in Schwarz–Christoffel Mapping